Shock Treatment: Heavy Quark
Energy Loss in a Novel Geometry
William Horowitz
The Ohio State University
April 3, 2009
With many thanks to Yuri Kovchegov and Ulrich Heinz
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pQCD Success in High-pT at RHIC:
(circa 2005)
Y. Akiba for the PHENIX collaboration,
hep-ex/0510008
– Consistency:
RAA(h)~RAA(p)
– Null Control:
RAA(g)~1
– GLV Calculation: Theory~Data for reasonable
fixed L~5 fm and dNg/dy~dNp/dy
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Trouble for High-pT wQGP Picture
p0 v2
– v2 too small
– NPE supp. too large
WHDG
C. Vale, QM09 Plenary (analysis by R. Wei)
NPE v2
STAR, Phys. Rev. Lett. 98, 192301 (2007)
Pert. at LHC energies?
PHENIX, Phys. Rev. Lett. 98, 172301 (2007)
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Motivation for High-pT AdS
• Why study AdS E-loss models?
– Many calculations vastly simpler
• Complicated in unusual ways
– Data difficult to reconcile with pQCD
– pQCD quasiparticle picture leads to
dominant q ~ m ~ .5 GeV mom. transfers
=> Nonperturbatively large as
• Use data to learn about E-loss
mechanism, plasma properties
– Domains of self-consistency crucial for
understanding
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Strong Coupling Calculation
• The supergravity double conjecture:
QCD  SYM  IIB
– IF super Yang-Mills (SYM) is not too
different from QCD, &
– IF Maldacena conjecture is true
– Then a tool exists to calculate stronglycoupled QCD in classical SUGRA
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AdS/CFT Energy Loss Models I
– Langevin Diffusion
• Collisional energy loss for heavy quarks
• Restricted to low pT
• pQCD vs. AdS/CFT computation of D, the
diffusion coefficient
Moore and Teaney, Phys.Rev.C71:064904,2005
Casalderrey-Solana and Teaney, Phys.Rev.D74:085012,2006; JHEP 0704:039,2007
– ASW/LRW model
• Radiative energy loss model for all parton species
• pQCD vs. AdS/CFT computation of
• Debate over its predicted magnitude
BDMPS, Nucl.Phys.B484:265-282,1997
Armesto, Salgado, and Wiedemann, Phys. Rev. D69 (2004) 114003
Liu, Ragagopal, Wiedemann, PRL 97:182301,2006; JHEP 0703:066,2007
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AdS/CFT Energy Loss Models II
String Drag calculation
–
–
–
–
–
Embed string rep. quark/gluon in AdS geom.
Includes all E-loss modes (difficult to interpret)
Gulotta, Pufu, Rocha, JHEP 0810:052, 2008
Gluons and light quarks Gubser,
Chesler, Jensen, Karch, Yaffe, arXiv:0810.1985 [hep-th]
Empty space HQ calculation Kharzeev, arXiv:0806.0358 [hep-ph]
Previous HQ: thermalized QGP plasma, temp. T,
Gubser, Phys.Rev.D74:126005,2006
Herzog, Karch, Kovtun, Kozcaz, Yaffe, JHEP 0607:013, 2006
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Energy Loss Comparison
D7 Probe Brane
t
z=0
– AdS/CFT Drag:
Q, m
zm = l1/2/2pm
dpT/dt ~ -(T2/Mq) pT
zh = 1/pT
v
x
3+1D Brane
Boundary
D3 Black Brane
(horizon)
Black Hole
z=
– Similar to Bethe-Heitler
dpT/dt ~ -(T3/Mq2) pT
– Very different from LPM
dpT/dt ~ -LT3 log(pT/Mq)
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LHC RcAA(pT)/RbAA(pT) Prediction
• Individual c and b RAA(pT) predictions:
WH and M. Gyulassy, Phys. Lett. B 666, 320 (2008)
– Taking the ratio cancels most normalization differences seen previously
– pQCD ratio asymptotically approaches 1, and more slowly so for increased
quenching (until quenching saturates)
– AdS/CFT ratio is flat and
many times smaller than pQCD at only moderate pT
WH and M. Gyulassy, Phys. Lett. B 666, 320 (2008)
– Distinguish rad and el contributions?
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Universality and Applicability
• How universal are th. HQ drag results?
– Examine different theories
– Investigate alternate geometries
• Other AdS geometries
– Bjorken expanding hydro
– Shock metric
• Warm-up to Bj. hydro
• Can represent both hot and cold nuclear matter
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New Geometries
Constant T Thermal Black Brane
Shock Geometries
J Friess, et al., PRD75:106003, 2007
Nucleus as Shock
DIS
Embedded String in Shock
Albacete, Kovchegov, Taliotis,
JHEP 0807, 074 (2008)
Before
After
vshock
Q
z
Bjorken-Expanding Medium
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x
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Q
z
vshock
x
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Standard Method of Attack
• Parameterize string worldsheet
m
– X (t, s)
• Plug into Nambu-Goto action
m
• Varying SNG yields EOM for X
• Canonical momentum flow (in t, s)
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New in the Shock
• Find string solutions in HQ rest frame
– vHQ = 0
• Assume static case (not new)
– Shock wave exists for all time
– String dragged for all time
m
• X = (t, x(z), 0,0, z)
• Simple analytic solutions:
– x(z) = x0, x0 ± m ½ z3/3
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Shock Geometry Results
• Three t-ind. solutions (static gauge):
m
X = (t, x(z), 0,0, z)
– x(z) = x0, x0 ± m ½ z3/3
Q
z=0
vshock
x0 + m ½ z3/3
x0 - m ½ z3/3
x0
x
z=
4/3/09
• Constant solution unstable
• Time-reversed negative x solution unphysical
• Sim. to x ~ z3/3, z << 1, for const. T BH geom.
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HQ Momentum Loss
x(z) = m ½ z3/3 =>
Relate m to nuclear properties
– Use AdS dictionary
• Metric in Fefferman-Graham form: m ~ T--/Nc2
– T’00 ~ Nc2 L4
• Nc2 gluons per nucleon in shock
• L is typical mom. scale; L-1 typical dist. scale
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Frame Dragging
• HQ Rest Frame
• Shock Rest Frame
Mq
vsh
L
vq = -vsh
1/L
i
vq = 0
i
Mq
vsh = 0
– Change coords, boost Tmn into HQ rest frame:
• T-- ~ Nc2 L4 g2 ~ Nc2 L4 (p’/M)2
• p’ ~ gM: HQ mom. in rest frame of shock
– Boost mom. loss into shock rest frame
– p0t = 0:
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Put Together
• This leads to
–Recall for BH:
–Shock gives exactly the same drag as BH for L = p T
• We’ve generalized the BH solution to
both cold and hot nuclear matter E-loss
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Shock Metric Speed Limit
• Local speed of light (in HQ rest frame)
– Demand reality of point-particle action
• Solve for v = 0 for finite mass HQ
– z = zM = l½/2pMq
– Same speed limit as for BH metric when L = pT
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Conclusions and Outlook
– Use data to test E-loss mechanism
• RcAA(pT)/RbAA(pT) wonderful tool
– Calculated HQ drag in shock geometry
• For L = p T, drag and speed limit identical to BH
• Generalizes HQ drag to hot and cold nuclear matter
– Unlike BH, quark mass unaffected by shock
• Quark always heavy from strong coupling dressing?
• BH thermal adjustment from plasma screening IR?
– Future work:
• Time-dependent shock treatment
• AdS E-loss in Bjorken expanding medium
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Backup Slides
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Canonical Momenta
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RAA Approximation
– Above a few GeV, quark production
spectrum is approximately power law:
• dN/dpT ~ 1/pT(n+1), where n(pT) has some
momentum dependence
y=0
RHIC
– We can approximate RAA(pT):
• RAA ~ (1-e(pT))n(pT),
where pf = (1-e)pi (i.e. e = 1-pf/pi)
LHC
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Looking for a Robust, Detectable Signal
– Use LHC’s large pT reach and identification of c
and b to distinguish between pQCD, AdS/CFT
• Asymptotic pQCD momentum loss:
erad ~ as L2 log(pT/Mq)/pT
• String theory drag momentum loss:
eST ~ 1 - Exp(-m L),
m = pl1/2 T2/2Mq
S. Gubser, Phys.Rev.D74:126005 (2006); C. Herzog et al. JHEP 0607:013,2006
– Independent of pT and strongly dependent on Mq!
– T2 dependence in exponent makes for a very sensitive probe
– Expect: epQCD
0 vs. eAdS indep of pT!!
• dRAA(pT)/dpT > 0 => pQCD; dRAA(pT)/dpT < 0 => ST
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Model Inputs
– AdS/CFT Drag: nontrivial mapping of QCD to SYM
• “Obvious”: as = aSYM = const., TSYM = TQCD
– D 2pT = 3 inspired: as = .05
– pQCD/Hydro inspired: as = .3 (D 2pT ~ 1)
• “Alternative”: l = 5.5, TSYM = TQCD/31/4
• Start loss at thermalization time t0; end loss at Tc
– WHDG convolved radiative and elastic energy loss
• as = .3
– WHDG radiative energy loss (similar to ASW)
•
= 40, 100
– Use realistic, diffuse medium with Bjorken expansion
– PHOBOS (dNg/dy = 1750); KLN model of CGC (dNg/dy = 2900)
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LHC c, b RAA pT Dependence
WH and M. Gyulassy,
Phys. Lett. B 666, 320 (2008)
– Significant
NaïvePrediction
LHC
Unfortunately,
Large
suppression
expectations
rise large
inZoo:
Rleads
met
suppression
What
(pTin
to
) for
full
flattening
a Mess!
pQCD
numerical
pQCD
Rad+El
similar
calculation:
to AdS/CFT
AA
– Use
Let’sofgorealistic
through
dRAA
geometry
step
(pT)/dp
by step
> 0 Bjorken
=> pQCD;
expansion
dRAA(pTallows
)/dpT <
saturation
0 => ST below .2
Tand
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An Enhanced Signal
• But what about the interplay between
mass and momentum?
– Take ratio of c to b RAA(pT)
• pQCD: Mass effects die out with increasing pT
RcbpQCD(pT) ~ 1 - as n(pT) L2 log(Mb/Mc) ( /pT)
– Ratio starts below 1, asymptotically approaches 1.
Approach is slower for higher quenching
• ST: drag independent of pT, inversely
proportional to mass. Simple analytic approx.
of uniform medium gives
RcbpQCD(pT) ~ nbMc/ncMb ~ Mc/Mb ~ .27
– Ratio starts below 1; independent of pT
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LHC RcAA(pT)/RbAA(pT) Prediction
• Recall the Zoo:
WH and M. Gyulassy, Phys. Lett. B 666, 320 (2008)
– Taking the ratio cancels most normalization differences seen previously
– pQCD ratio asymptotically approaches 1, and more slowly so for increased
quenching (until quenching saturates)
– AdS/CFT ratio is flat and
many times smaller than pQCD at only moderate pT
WH and M. Gyulassy, Phys. Lett. B 666, 320 (2008)
– Distinguish rad and el contributions?
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Additional Discerning Power
– Consider ratio for ALICE pT reach
mc = mb = 0
– Adil-Vitev in-medium fragmentation rapidly approaches, and then broaches, 1
» Does not include partonic E-loss, which will be nonnegligable as ratio goes to unity
– Higgs (non)mechanism => Rc/Rb ~ 1 ind. of pT
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Not So Fast!
• Speed limit estimate for
applicability of AdS drag
D7 Probe Brane
Q
Worldsheet boundary
Spacelike if g > gcrit
– g < gcrit = (1 + 2Mq/l1/2 T)2
~ 4Mq2/(l T2)
z
Trailing
String
“Brachistochrone”
• Limited by Mcharm ~ 1.2 GeV
• Similar to BH LPM
– gcrit ~ Mq/(lT)
x
• No single T for QGP
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D3 Black Brane
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LHC RcAA(pT)/RbAA(pT) Prediction
(with speed limits)
WH and M. Gyulassy, Phys. Lett. B 666, 320 (2008)
– T(t0): (, highest T—corrections unlikely for smaller momenta
– Tc: ], lowest T—corrections likely for higher momenta
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Derivation of BH Speed Limit I
• Constant HQ velocity
– Assume const. v kept by F.v
Minkowski Boundary
z=0
2
½
– Critical field strength Ec = M /l
• E > Ec: Schwinger pair prod. zM =
• Limits g < gc ~ T2/lM2
l½ /
2pM
E
F.v = dp/dt
Q v
D7
dp/dt
J. Casalderrey-Solana and D. Teaney, JHEP 0704, 039 (2007)
– Alleviated by allowing var. v
• Drag similar to const. v
Herzog, Karch, Kovtun, Kozcaz, Yaffe, JHEP 0607:013 (2006)
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zh = 1/pT
D3
z=
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Derivation of BH Speed Limit II
• Local speed of light
– BH Metric => varies with depth z
• v(z)2 < 1 – (z/zh)4
l½/2pM
– HQ located at zM =
– Limits g < gc ~ T2/lM2
• Same limit as from const. v
S. S. Gubser, Nucl. Phys. B 790, 175 (2008)
zM =
Minkowski Boundary
z=0
l½ /
2pM
– Mass a strange beast
• Mtherm < Mrest
• Mrest  Mkin
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D7
dp/dt
zh = 1/pT
– Note that M >> T
E
F.v = dp/dt
Q v
D3
z=
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Trouble for High-pT wQGP Picture
p0 v2
– v2 too small
– NPE supp. too large
WHDG dN/dy = 1400
C. Vale, QM09 Plenary (analysis by R. Wei)
NPE v2
STAR, Phys. Rev. Lett. 98, 192301 (2007)
Pert. at LHC energies?
PHENIX, Phys. Rev. Lett. 98, 172301 (2007)
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Measurement at RHIC
– Future detector upgrades will allow for identified c
and b quark measurements
– RHIC production spectrum significantly
harder than LHC
•
• NOT slowly varying
y=0
RHIC
– No longer expect
pQCD dRAA/dpT > 0
• Large n requires
corrections to naïve
Rcb ~ Mc/Mb
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LHC
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RHIC c, b RAA pT Dependence
WH, M. Gyulassy, arXiv:0710.0703 [nucl-th]
• Large increase in n(pT) overcomes reduction in
E-loss and makes pQCD dRAA/dpT < 0, as well
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RHIC Rcb Ratio
pQCD
pQCD
AdS/CFT
AdS/CFT
WH, M. Gyulassy, arXiv:0710.0703 [nucl-th]
• Wider distribution of AdS/CFT curves due to large n:
increased sensitivity to input parameters
• Advantage of RHIC: lower T => higher AdS speed limits
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HQ Momentum Loss in the Shock
x(z) = m ½ z3/3 =>
• Must boost into shock rest frame:
• Relate m to nuclear properties
– Use AdS dictionary
• Metric in Fefferman-Graham form: m ~ T--/Nc2
– T00 ~ Nc2 L4
• Nc2 gluons per nucleon in shock
• L is typical mom. scale; L-1 typical dist. Scale
– Change coords, boost into HQ rest frame:
• T-- ~ Nc2 L4 (p/M)2
=> m = L4 (p/M)2
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HQ Momentum Loss in the Shock
x(z) = m ½ z3/3 =>
Relate m to nuclear properties
– Use AdS dictionary: m ~ T--/Nc2
– T-- = (boosted den. of scatterers) x (mom.)
– T-- = Nc2 (L3 p+/L) x (p+)
•
•
•
•
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Nc2 gluons per nucleon in shock
L is typical mom. scale; L-1 typical dist. scale
p+: mom. of shock gluons as seen by HQ
p: mom. of HQ as seen by shock
=> m = L2p+2
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HQ Drag in the Shock
• HQ Rest Frame
• Shock Rest Frame
Mq
vsh
L
vq = -vsh
1/L
i
vq = 0
i
Mq
vsh = 0
–Recall for BH:
–Shock gives exactly the same drag as BH for L = p T
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HQ Momentum Loss
x(z) = m ½ z3/3 =>
Relate m to nuclear properties
– Use AdS dictionary
• Metric in Fefferman-Graham form: m ~ T--/Nc2
– T’00 ~ Nc2 L4
• Nc2 gluons per nucleon in shock
• L is typical mom. scale; L-1 typical dist. scale
– Change coords, boost into HQ rest frame:
• T-- ~ Nc2 L4 g2 ~ Nc2 L4 (p’/M)2
• p’ ~ gM: HQ mom. in rest frame of shock
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Shocking Drag
• HQ Rest Frame
• Shock Rest Frame
Mq
vsh
L
vq = -vsh
1/L
vq = 0
i
i
Mq
vsh = 0
• Boost mom. loss into shock rest frame
– p0t = 0:
• Therefore
–Recall for BH:
–Shock gives exactly the same drag as BH for L = p T
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